Reverse-order fourth term $3b-2a$ creates a node conflict

A tree witness is a function $\theta(k,r)\in\{0,1\}$ on the nodes $(k,r)$, $0\le r<2^k$, of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ (two's-complement digits). A finite word $Q$ of integers is $\theta$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression; it is tree-safe if it is $\theta$-safe for some $\theta$. Let $P$ be a finite word and $a\neq b$ letters of $P$ with $b$ occurring before $a$. Put $c=3b-2a$ and suppose

Then the word $P\cdot c$ is not tree-safe. Indeed the in-order pairs $(b,a)$ and $(a,c)$ are then both open (neither continuation present), they sit at the same tree node $(k,\,a\bmod 2^k)$ with $k=\nu_2(b-a)$, and they force opposite bits there, since $\mathrm{bit}_k(c)=\mathrm{bit}_k(b)\neq\mathrm{bit}_k(a)$. This explains why the isolated-minimum completion by the fourth term $3b-2a$ has no analogue when the pair is taken in reverse order (Erdős Problem 195, tree-witness approach).

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. Positions in a word are List.idxOf; P ++ [c] appends c.